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Properties

Label 24.48.1.be.1
Level $24$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $2$

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Invariants

Level: $24$ $\SL_2$-level: $8$ Newform level: $288$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $2$ are rational) Cusp widths $4^{4}\cdot8^{4}$ Cusp orbits $1^{2}\cdot2^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8G1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.48.1.85

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&20\\0&13\end{bmatrix}$, $\begin{bmatrix}3&2\\8&7\end{bmatrix}$, $\begin{bmatrix}7&10\\0&5\end{bmatrix}$, $\begin{bmatrix}11&20\\4&23\end{bmatrix}$, $\begin{bmatrix}13&6\\0&5\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.be.1.1, 24.96.1-24.be.1.2, 24.96.1-24.be.1.3, 24.96.1-24.be.1.4, 24.96.1-24.be.1.5, 24.96.1-24.be.1.6, 24.96.1-24.be.1.7, 24.96.1-24.be.1.8, 24.96.1-24.be.1.9, 24.96.1-24.be.1.10, 24.96.1-24.be.1.11, 24.96.1-24.be.1.12, 24.96.1-24.be.1.13, 24.96.1-24.be.1.14, 24.96.1-24.be.1.15, 24.96.1-24.be.1.16, 120.96.1-24.be.1.1, 120.96.1-24.be.1.2, 120.96.1-24.be.1.3, 120.96.1-24.be.1.4, 120.96.1-24.be.1.5, 120.96.1-24.be.1.6, 120.96.1-24.be.1.7, 120.96.1-24.be.1.8, 120.96.1-24.be.1.9, 120.96.1-24.be.1.10, 120.96.1-24.be.1.11, 120.96.1-24.be.1.12, 120.96.1-24.be.1.13, 120.96.1-24.be.1.14, 120.96.1-24.be.1.15, 120.96.1-24.be.1.16, 168.96.1-24.be.1.1, 168.96.1-24.be.1.2, 168.96.1-24.be.1.3, 168.96.1-24.be.1.4, 168.96.1-24.be.1.5, 168.96.1-24.be.1.6, 168.96.1-24.be.1.7, 168.96.1-24.be.1.8, 168.96.1-24.be.1.9, 168.96.1-24.be.1.10, 168.96.1-24.be.1.11, 168.96.1-24.be.1.12, 168.96.1-24.be.1.13, 168.96.1-24.be.1.14, 168.96.1-24.be.1.15, 168.96.1-24.be.1.16, 264.96.1-24.be.1.1, 264.96.1-24.be.1.2, 264.96.1-24.be.1.3, 264.96.1-24.be.1.4, 264.96.1-24.be.1.5, 264.96.1-24.be.1.6, 264.96.1-24.be.1.7, 264.96.1-24.be.1.8, 264.96.1-24.be.1.9, 264.96.1-24.be.1.10, 264.96.1-24.be.1.11, 264.96.1-24.be.1.12, 264.96.1-24.be.1.13, 264.96.1-24.be.1.14, 264.96.1-24.be.1.15, 264.96.1-24.be.1.16, 312.96.1-24.be.1.1, 312.96.1-24.be.1.2, 312.96.1-24.be.1.3, 312.96.1-24.be.1.4, 312.96.1-24.be.1.5, 312.96.1-24.be.1.6, 312.96.1-24.be.1.7, 312.96.1-24.be.1.8, 312.96.1-24.be.1.9, 312.96.1-24.be.1.10, 312.96.1-24.be.1.11, 312.96.1-24.be.1.12, 312.96.1-24.be.1.13, 312.96.1-24.be.1.14, 312.96.1-24.be.1.15, 312.96.1-24.be.1.16
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $1536$

Jacobian

Conductor: $2^{5}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 288.2.a.d

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 99x + 378 $
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Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(0:1:0)$, $(6:0:1)$

Maps to other modular curves

$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{3^4}\cdot\frac{72x^{2}y^{14}-1806786x^{2}y^{12}z^{2}+11992510728x^{2}y^{10}z^{4}-41724737243361x^{2}y^{8}z^{6}+86316768541603008x^{2}y^{6}z^{8}-108430149204232868169x^{2}y^{4}z^{10}+77683977836450211140028x^{2}y^{2}z^{12}-24540171192307636348712985x^{2}z^{14}-2844xy^{14}z+39791736xy^{12}z^{3}-220919885919xy^{10}z^{5}+678690724017294xy^{8}z^{7}-1276500825658236744xy^{6}z^{9}+1474784629965103265208xy^{4}z^{11}-974922954834855543091497xy^{2}z^{13}+281850771115636280944373610xz^{15}-y^{16}+76464y^{14}z^{2}-653466852y^{12}z^{4}+2746648173672y^{10}z^{6}-6688178273963052y^{8}z^{8}+10054525978802747232y^{6}z^{10}-9117545922371977768746y^{4}z^{12}+4456390437125047413495792y^{2}z^{14}-807658463770743059542110681z^{16}}{z^{2}y^{4}(x^{2}y^{8}-76572x^{2}y^{6}z^{2}+458909145x^{2}y^{4}z^{4}-746313854976x^{2}y^{2}z^{6}+351689661333504x^{2}z^{8}-72xy^{8}z+1759401xy^{6}z^{3}-7322001642xy^{4}z^{5}+9756822159360xy^{2}z^{7}-4039254716940288xz^{9}+2754y^{8}z^{2}-28083024y^{6}z^{4}+66217135257y^{4}z^{6}-51787102961664y^{2}z^{8}+11574700493635584z^{10})}$

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.24.0.e.2 $8$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.h.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.1.c.1 $24$ $2$ $2$ $1$ $0$ dimension zero

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.96.1.h.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.x.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bd.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bh.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bv.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bz.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.cb.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.cd.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.144.9.gv.1 $24$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
24.192.9.dz.1 $24$ $4$ $4$ $9$ $1$ $1^{4}\cdot2^{2}$
120.96.1.fx.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.gb.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.gn.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.gr.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ij.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.in.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.iz.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.jd.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.240.17.dh.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.ezm.1 $120$ $6$ $6$ $17$ $?$ not computed
168.96.1.fx.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.gb.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.gn.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.gr.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.ij.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.in.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.iz.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.jd.2 $168$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.fx.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.gb.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.gn.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.gr.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.ij.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.in.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.iz.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.jd.2 $264$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.fx.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.gb.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.gn.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.gr.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.ij.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.in.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.iz.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.jd.2 $312$ $2$ $2$ $1$ $?$ dimension zero